Gaussian Variational Approximation with Composite Likelihood for Crossed Random Effect Models

Composite likelihood usually ignores dependencies among response components, while variational approximation to likelihood ignores dependencies among parameter components.What both methods have in common is that they essentially break the dependence of random effects.In this paper, we derive a Gaussian variational approximation to the composite log-likelihood function for Poisson and Gamma models with crossed random effects.We present theoretical aspects of the estimates derived from this approximation and support these theories with simulation studies.Specifically, we show the estimates are consistent with a convergence rate m -1/2 +n -1/2 , where m and n denote the number of rows and columns, respectively.We further provide detailed asymptotic normality results under a new regime where log m/ log n for (1/2, 2).Additional simulation studies show that our method yields comparable estimation performance and is slightly faster than the Laplace approximation in the package glmmTMB and a Gaussian variational approximation to the full log-likelihood function.